Error Ellipses

From February 1999, the errors for the position and the proper motions are expressed as error ellipses, made of three numbers: the major axis, the minor axis, and the position angle of the major axis (the position angle is measured from the North to the East).

The units in which the major/minor axises are expressed are mas for the position, and mas/yr for the proper motion; the position angle is expressed in degrees, in the range [0°,180°[

If we designate by (a,b,p) the 3 values of major axis, minor axis, position angle, the variances and covariance on the parameters A (α.cos(δ) or µ_α) and D (δ or µ_δ) are:

   var(A) = sin²(p) a²+ cos²(p) b²

   var(D) = cos²(p) a²+ sin²(p) b²

   cov(A,D) = (a²– b²) cos(p) sin(p)

and from there the mean errors and correlation coefficients may be computed knowning that:

   σ(A) = sqrt(var(A))

   σ(D) = sqrt(var(D))

   ρ = cov(A,D)/(σ(A)⋅σ(D))

	var(A)	= sin²(p) a²+ cos²(p) b²
	var(D)	= cos²(p) a²+ sin²(p) b²
	cov(A,D)	= (a²– b²) cos(p) sin(p)

	σ(A)	= sqrt(var(A))
	σ(D)	= sqrt(var(D))
	ρ	= cov(A,D)/(σ(A)⋅σ(D))